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Your have probably to work with mouse coordinates and with three events :

onmousedown : 1/- store the mouse initial coordinates, 2/- store them in old coordinates and 3/- set a boolean okMov to true
onmousemouse : if (okMov) {1/- store the mouse current coordinates, 2/- rotate the cube with current-old ccordinates and 3/- store current coordinates in old coordinates}
onmouseup : okMov=false;

The mouse coordinates and the boolean okMov are obviously global variables...
See this simple page cube.htm (line 84 to 113 - we also define touchstart, touchmove and touchrend) for the principles. You have only to change the move ...